An%20Introduction%20to%20Logistic%20Regression

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An Introduction to Logistic Regression

• JohnWhitehead
• Department of Economics
• Appalachian State University

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Outline

• Introduction and Description
• Some Potential Problems and Solutions
• Writing Up the Results

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Introduction and Description

• Why use logistic regression?
• Estimation by maximum likelihood
• Interpreting coefficients
• Hypothesis testing
• Evaluating the performance of the model

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Why use logistic regression?

• There are many important research topics for
  which the dependent variable is "limited."
• For example voting, morbidity or mortality, and
  participation data is not continuous or
  distributed normally.
• Binary logistic regression is a type of
  regression analysis where the dependent variable
  is a dummy variable coded 0 (did not vote) or
  1(did vote)

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The Linear Probability Model

• In the OLS regression
• Y ? ??X e where Y (0, 1)
• The error terms are heteroskedastic
• e is not normally distributed because Y takes on
  only two values
• The predicted probabilities can be greater than 1
  or less than 0

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An Example Hurricane Evacuations
Q EVAC Did you evacuate your home to go
someplace safer before Hurricane Dennis (Floyd)
hit?   1 YES 2 NO 3 DON'T KNOW 4 REFUSED
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The Data
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OLS Results
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Problems
Predicted Values outside the 0,1 range
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Heteroskedasticity
Park Test
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The Logistic Regression Model

• The "logit" model solves these problemslnp/(1-
  p) ?? ?X e
• p is the probability that the event Y occurs,
  p(Y1)
• p/(1-p) is the "odds ratio"
• lnp/(1-p) is the log odds ratio, or "logit"

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• More
• The logistic distribution constrains the
  estimated probabilities to lie between 0 and 1.
• The estimated probability is p 1/1 exp(-?
  - ? X)
• if you let ? ? X 0, then p .50
• as ? ? X gets really big, p approaches 1
• as ? ? X gets really small, p approaches 0

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(No Transcript)
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Comparing LP and Logit Models
Y
LP Model
1
X
0
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Maximum Likelihood Estimation (MLE)

• MLE is a statistical method for estimating the
  coefficients of a model.
• The likelihood function (L) measures the
  probability of observing the particular set of
  dependent variable values (p1, p2, ..., pn) that
  occur in the sample L Prob (p1 p2 pn)
• The higher the L, the higher the probability of
  observing the ps in the sample.

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• MLE involves finding the coefficients (?, ?) that
  makes the log of the likelihood function (LL lt 0)
  as large as possible
• Or, finds the coefficients that make -2 times the
  log of the likelihood function (-2LL) as small as
  possible
• The maximum likelihood estimates solve the
  following condition Y - p(Y1)Xi
  0summed over all observations, i 1,,n

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Interpreting Coefficients

• Since lnp/(1-p) ?? ?X e
• The slope coefficient (?) is interpreted as the
  rate of change in the "log odds" as X changes
  not very useful.
• Since p 1/1 exp(-? - ? X) The marginal
  effect of a change in X on the probability is
  ?p/?X f(? X) ?

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• An interpretation of the logit coefficient which
  is usually more intuitive is the "odds ratio"
• Since p/(1-p) exp(? ?X)
• exp(?) is the effect of the independent variable
  on the "odds ratio"

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From SPSS Output
Households with pets are 1.933 times more likely
to evacuate than those without pets.
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Hypothesis Testing

• The Wald statistic for the ? coefficient
  is Wald ? /s.e.B2which is distributed
  chi-square with 1 degree of freedom.
• The "Partial R" (in SPSS output) is R
  (Wald-2)/(-2LL(?)1/2

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An Example
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Evaluating the Performance of the Model

• There are several statistics which can be used
  for comparing alternative models or evaluating
  the performance of a single model
• Model Chi-Square
• Percent Correct Predictions
• Pseudo-R2

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Model Chi-Square

• The model likelihood ratio (LR), statistic is
  LRi -2LL(?) - LL(?, ?) Or, as you are
  reading SPSS printout LRi -2LL (of
  beginning model) - -2LL (of ending model)
• The LR statistic is distributed chi-square with i
  degrees of freedom, where i is the number of
  independent variables
• Use the Model Chi-Square statistic to determine
  if the overall model is statistically
  significant.

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An Example
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Percent Correct Predictions

• The "Percent Correct Predictions" statistic
  assumes that if the estimated p is greater than
  or equal to .5 then the event is expected to
  occur and not occur otherwise.
• By assigning these probabilities 0s and 1s and
  comparing these to the actual 0s and 1s, the
  correct Yes, correct No, and overall correct
  scores are calculated.

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An Example
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Pseudo-R2

• One psuedo-R2 statistic is the McFadden's-R2
  statistic McFadden's-R2 1 -
  LL(?,?)/LL(?) 1 - -2LL(?, ?)/-2LL(?)
  (from SPSS printout)
• where the R2 is a scalar measure which varies
  between 0 and (somewhat close to) 1 much like the
  R2 in a LP model.

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An Example
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Some potential problems and solutions

• Omitted Variable Bias
• Irrelevant Variable Bias
• Functional Form
• Multicollinearity
• Structural Breaks

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Omitted Variable Bias

• Omitted variable(s) can result in bias in the
  coefficient estimates. To test for omitted
  variables you can conduct a likelihood ratio
  testLRq -2LL(constrained model, ik-q)
  - -2LL(unconstrained model, ik) where LR
  is distributed chi-square with q degrees of
  freedom, with q 1 or more omitted variables
• This test is conducted automatically by SPSS if
  you specify "blocks" of independent variables

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An Example
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Constructing the LR Test
Since the chi-squared value is less than the
critical value the set of coefficients is not
statistically significant. The full model is not
an improvement over the partial model.
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Irrelevant Variable Bias

• The inclusion of irrelevant variable(s) can
  result in poor model fit.
• You can consult your Wald statistics or conduct a
  likelihood ratio test.

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Functional Form

• Errors in functional form can result in biased
  coefficient estimates and poor model fit.
• You should try different functional forms by
  logging the independent variables, adding squared
  terms, etc.
• Then consult the Wald statistics and model
  chi-square statistics to determine which model
  performs best.

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Multicollinearity

• The presence of multicollinearity will not lead
  to biased coefficients.
• But the standard errors of the coefficients will
  be inflated.
• If a variable which you think should be
  statistically significant is not, consult the
  correlation coefficients.
• If two variables are correlated at a rate greater
  than .6, .7, .8, etc. then try dropping the least
  theoretically important of the two.

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Structural Breaks

• You may have structural breaks in your data.
  Pooling the data imposes the restriction that an
  independent variable has the same effect on the
  dependent variable for different groups of data
  when the opposite may be true.
• You can conduct a likelihood ratio testLRi1
  -2LL(pooled model) -2LL(sample 1)
  -2LL(sample 2) where samples 1 and 2 are
  pooled, and i is the number of independent
  variables.

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An Example

• Is the evacuation behavior from Hurricanes Dennis
  and Floyd statistically equivalent?

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Constructing the LR Test
Since the chi-squared value is greater than the
critical value the set of coefficients are
statistically different. The pooled model is
inappropriate.
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What should you do?

• Try adding a dummy variableFLOYD 1 if Floyd,
  0 if Dennis

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Writing Up Results

• Present descriptive statistics in a table
• Make it clear that the dependent variable is
  discrete (0, 1) and not continuous and that you
  will use logistic regression.
• Logistic regression is a standard statistical
  procedure so you don't (necessarily) need to
  write out the formula for it. You also (usually)
  don't need to justify that you are using Logit
  instead of the LP model or Probit (similar to
  logit but based on the normal distribution the
  tails are less fat).

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An Example
"The dependent variable which measures the
willingness to evacuate is EVAC. EVAC is equal to
1 if the respondent evacuated their home during
Hurricanes Floyd and Dennis and 0 otherwise. The
logistic regression model is used to estimate the
factors which influence evacuation behavior."
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Organize your regression results in a table

• In the heading state that your dependent variable
  (dependent variable EVAC) and that these are
  "logistic regression results.
• Present coefficient estimates, t-statistics (or
  Wald, whichever you prefer), and (at least the)
  model chi-square statistic for overall model fit
• If you are comparing several model specifications
  you should also present the correct predictions
  and/or Pseudo-R2 statistics to evaluate model
  performance
• If you are comparing models with hypotheses about
  different blocks of coefficients or testing for
  structural breaks in the data, you could present
  the ending log-likelihood values.

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An Example
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When describing the statistics in the tables,
point out the highlights for the reader. What
are the statistically significant variables?

• "The results from Model 1 indicate that coastal
  residents behave according to risk theory. The
  coefficient on the MOBLHOME variable is negative
  and statistically significant at the p lt .01
  level (t-value 5.42). Mobile home residents are
  4.75 times more likely to evacuate.

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Is the overall model statistically significant?
The overall model is significant at the .01
level according to the Model chi-square
statistic. The model predicts 69.5 of the
responses correctly. The McFadden's R2 is .066."
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Which model is preferred?
"Model 2 includes three additional independent
variables. According to the likelihood ratio test
statistic, the partial model is superior to the
full model of overall model fit. The block
chi-square statistic is not statistically
significant at the .01 level (critical value
11.35 df3). The coefficient on the children,
gender, and race variables are not statistically
significant at standard levels."
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Also

• You usually don't need to discuss the magnitude
  of the coefficients--just the sign ( or -) and
  statistical significance.
• If your audience is unfamiliar with the
  extensions (beyond SPSS or SAS printouts) to
  logistic regression, discuss the calculation of
  the statistics in an appendix or footnote or
  provide a citation.
• Always state the degrees of freedom for your
  likelihood-ratio (chi-square) test.

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References

• http//personal.ecu.edu/whiteheadj/data/logit/
• http//personal.ecu.edu/whiteheadj/data/logit/logi
  tpap.htm
• E-mail WhiteheadJ_at_mail.ecu.edu